What's the secret of high frequency harmonics?

[tulamide]

The question is very interesting. Let's set the scene first. We all know that sounds have more or less harmonics and that different waveforms result from the mixture of harmonics and levels.

The third harmonic will have 3x the frequency of the fundamental (which is the first harmonic). Now let's assume we want to generate a sawtooth. The 3rd harmonic will also have 1/3rd the level of the fundamental. I can continue bulding my waveform with these simple rules.

But, if I want a perfect sawtooth while playing at 220 Hz, I would need to add all harmonics of the audible range. I did the math for you.

That's a total of 100 harmonics if we assume 22.05 kHz to mark the upper end of the audible range (220, 440, 660, 880, etc.).

Ok. But now I play at 1600 Hz.

This time it's only 13 harmonics (1600, 3200, 4800, 6400, etc.).

Yet it's still a sawtooth and clearly recognizable as such. Why is that? We lose information, but the waveform stays intact. I don't understand it.

[Logado]

try saw waveform and see how harmonics build saw
https://bl.ocks.org/jinroh/7524988
If we play a note close to Nyquist Frequency (sample rate / 2) then we need only one harmonic (sine)

[KG_is_back]

The simple explanation is that your brain fills in the blank. Or more precisely, it notices the common pattern in both and based on that your mind puts both of them "in the same box" in your mind. It may even depend on context. If you think about it, all sounds above cca. 10kHz should sound the same, because the only thing you hear is the fundamental (all harmonics are gone). The concept of a sawtooth sound is not frequency-dependent - it only depends on relationships between harmonics that are there (resp. are expected to be there). In fact, it is also not phase-dependent - you can randomize the phases of the individual harmonics and it will still sound the same, despite the fact the shape of the wave will be very different.

[MyCo]

It's just one of those interresting properties of human pattern recognition... you can exploit this property and build huge wavetables with it, like serum and wave codex do with very minimal memory requirements.

I've done similar calculations to see how small you can get a wavetable without loosing harmonic content. FS uses internally a 1024 wave X 2048 samples wavetable which is 8MB per wave type. According to the attached excel sheet you can get the same results with barely noticable errors with a 10 waves X 2048 samples table. So you could fit ~100 different waves into the same 8MB space.
I experimented with it, but didn't go any further than building test OSC.

[tulamide]

Wow that's even more interesting than I thought! It is dependend on perception? Does it belong to the field of "psychoacoustics"?

Somehow reminds of the basis for the mp3 codec (omitting information that is technically overwritten by other information and therefore only noticed via perception).

[110]

first of all, i dont know where to place this:

We all know that sounds have more or less harmonics and that different waveforms result from the mixture of harmonics and levels.

there are also sounds which are made by pink noise, fft´ed an d split onto 6 channels, each multiplyed with a random number greater than pi.

there are also sounds create by throwing an upright piano down the stars.

and there are probabyl 15+ different methods of creating a sawtooth wave in a programming language, summing odd harmonics is only one of them.

now to your question. when i am not mistaken the answer is as simple as it could be: if the base frequency is already 6Khz 13 harmonics are enough becase 1.) your samplingrate will not be able to transport higher frequencies than that, and 2.) your ears wouldnt be able to recognize it.

btw, there are several other interesting phenomena like this, including air absorbtion anomalities, the mystery of phase tracking (by humans) but also simple things like that fact that you can not track the direction sub bass is coming from, or that a a pieee of music seems to be faster and higher pitched when you turn up the volume.

[tulamide]

first of all, i dont know where to place this:

We all know that sounds have more or less harmonics and that different waveforms result from the mixture of harmonics and levels.

there are also sounds which are made by pink noise, fft´ed an d split onto 6 channels, each multiplyed with a random number greater than pi.

there are also sounds create by throwing an upright piano down the stars.

and there are probabyl 15+ different methods of creating a sawtooth wave in a programming language, summing odd harmonics is only one of them.

now to your question. when i am not mistaken the answer is as simple as it could be: if the base frequency is already 6Khz 13 harmonics are enough becase 1.) your samplingrate will not be able to transport higher frequencies than that, and 2.) your ears wouldnt be able to recognize it.

btw, there are several other interesting phenomena like this, including air absorbtion anomalities, the mystery of phase tracking (by humans) but also simple things like that fact that you can not track the direction sub bass is coming from, or that a a pieee of music seems to be faster and higher pitched when you turn up the volume.

This a very strange post. I'm not sure that you understand what sound is? The first 3 paragraphs of your answer don't debunk my words. For example, what do you think noise is? You should first be able to answer the question about sound before trying this one.

You describe 3 different methods of producing a sound. I was talking about the sound itself, when it exists. How you get to there is not the topic of my words.

And repeating what others already said is not really contributing to such an interesting topic.

Btw, you never get a sawtooth by summing odd harmonics.

[rocknrollkat]

The simple explanation is that your brain fills in the blank. Or more precisely, it notices the common pattern in both and based on that your mind puts both of them "in the same box" in your mind. It may even depend on context. If you think about it, all sounds above cca. 10kHz should sound the same, because the only thing you hear is the fundamental (all harmonics are gone). The concept of a sawtooth sound is not frequency-dependent - it only depends on relationships between harmonics that are there (resp. are expected to be there). In fact, it is also not phase-dependent - you can randomize the phases of the individual harmonics and it will still sound the same, despite the fact the shape of the wave will be very different.

....If you think about it, all sounds above cca. 10kHz should sound the same, because the only thing you hear is the fundamental (all harmonics are gone)....

This should be true, but here's the rub.
I was talking with Rupert Neve at the A.E.S. show about a decade ago and this question popped up.
He had built a test rig that was switchable between sine and square waves at over 10 kHz., just as you suggested.
To his amazement, he could hear the different timbres in the two signals !
I recreated his test in Flowstone, and yes, there is an audible difference at 10 kHz !
See for yourself !

ROXY

[Spogg]

I checked this with the spectrum analyser and the square wave has a lower and higher harmonic than the fundamental in addition, but not the sine.

My ears can hear the lower harmonic cut in with the square wave selected.

Martin’s oscillators are brilliant, but at this frequency I think there’s a wee bit of aliasing.

Schematic with the analyser attached.

Cheers

Spogg

[RJHollins]

interesting.

Aside. Spogg ... where did you get the A-weight module?